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Domain perturbations, capacity and shift of eigenvalues. (English) Zbl 1026.47501
Journées “Équations aux dérivées partielles”, Saint-Jean-de-Monts, France, 31 mai au 4 juin 1999. Exposés Nos. I–XIX. Nantes: Université de Nantes. Exp. No. VIII, 10 p. (1999).
Summary: After introducing the notion of capacity in a general Hilbert space setting we look at the spectral bound of an arbitrary self-adjoint and semi-bounded operator \(H\). If \(H\) is subjected to a domain perturbation, the spectrum is shifted to the right. We show that the magnitude of this shift can be estimated in terms of the capacity. We improve the upper bound on the shift which was given in [A. Noll, Commun. Partial Differ. Equations 24, 759–775 (1999; Zbl 0929.47023)] and obtain a lower bound which leads to a generalization of Thirring’s inequality if the underlying Hilbert space is an \(L^2\)-space. Moreover, a similar capacity upper bound for the second eigenvalue is established. The results are finally applied to higher-order partial differential operators.
For the entire collection see [Zbl 0990.00047].
MSC:
47A55 Perturbation theory of linear operators
31C15 Potentials and capacities on other spaces
35P05 General topics in linear spectral theory for PDEs
47B25 Linear symmetric and selfadjoint operators (unbounded)
47F05 General theory of partial differential operators (should also be assigned at least one other classification number in Section 47-XX)
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